The sampling distribution is the distribution of the values of a sample statistic computed for each possible sample that could be drawn from the target population under a specified sampling plan.2 Suppose a simple random sample of five hospitals is to be drawn from a population of 20 hospitals, There are (20 x 19 x 18 x 17 x 16)/(1 x 2 x 3 x 4 x 5). or 15.504 different.
samples of size 5 that can be drawn. The relative frequency distribution of the values of the mean of these 15,504 different samples would specify the sampling distribution of the mean.
An important task in marketing research is to calculate statistics, such as the sample mean and sample proportion, and use them to estimate the corresponding true population values, This process of generalizing the sample results to the population results is referred to as statistical inference, In practice, a single sample of predetermined size is selected and the sample statistics (such as mean and proportion) are computed. Hypothetically, in order to estimate the population parameter from the sample statistic, every possible sample that could have been drawn should be examined, If all possible samples were actually to be drawn, the distribution of the statistic would be the sampling distribution, Although in practice only one sample is actually drawn, the concept of a sampling distribution is still relevant, It enables us to use probability theory to make inferences about the-population values, The important properties of the sampling distribution of the mean, and the corresponding properties for the proportion, for large samples (30 or more) are as follows:
- The sampling distribution of the mean is ” normal distribution Strictly speaking the sampling distribution of a proportion is a binomial. However, for large samples (n =10 or more), it can be approximated by the normal distribution.
- The mean of the sampling distribution of the mean or the proportion.
- The standard deviation is called the standard error of the mean or the proportion to indicate that it refers to a sampling distribution of the mean or the proportion, and not to a sample or a population. The formulas are:
- Often the population standard deviation a is not known In these cases, it can be estimated from the sample by using the following formula: or In cases where a is estimated by s, the standard error of the mean becomes Assuming no measurement error, the reliability of an estimate of a population parameter can be assessed in terms of its standard error.
- Likewise, the standard error of the proportion can be estimated by using the sample proportion p as an estimator of the population proportion, π,as:
- The area under the sampling distribution between any two points can be calculated in terms of z values, The z value for a point is the number of standard errors a point is away from the mean or proportion, The z values may be computed as follows: and For example, the areas under one side of the curve between the mean and points that have z values of 1.0,2.0, and 3.0 are, respectively, 0.3413, 0.4772, and 0.4986.
- When the sample size is 10 percent or more of the population size, the standard error formulas will overestimate the standard deviation of the population mean or proportion, Hence, these should be adjusted by a finite population correction factor defined by: In this case and